Poisson Process for 500 variates

I would like to generate a code to produce a poisson process for N = 500 variates witk k is randomly between 1 and 10 and lambda = 12.

N = 500;

T = 1;

lambda = 12;

t  = linspace(0, T, 100);

dt = t(2) - t(1);
  
S=cumsum([zeros(1,N); poissrnd(lambda*dt, length(t)-1, N) ]);

plot(t,S)

I generated this code by myself, bu I cannot use the random value of k between 1 and 10 , and 500-variates graph has too many lines, but I guess this result is totally wrong, how can I get rid of my mistakes and write the code properly?

Thanks a lot.

My second code is this

T = 1;

lambda = 12;

k=9;

t  = linspace(0, T, 500);

for i = 1:500
    
f(i) = (lambda.^k) .* exp(-lambda) ./ factorial(k);
  
end

N = cumsum(f);

disp(N)

stairs(t,N(1:100))

But, again, the graph and estimations are totally wrong.

Answer 1

[Abdullah Gül]

Hello Zeynep,

You may try this.

lambda = 12;
nPeriods = 500;
dt = 0.1;   %time steps
T = nPeriods*dt;
t = 0:0.1:T;
rng('default')
k=randi([1,10],1,nPeriods);
f = (lambda.^k).*exp(-lambda)./factorial(k); % Poission Dist. 
Nd = cumsum(f);
N = [ 0 Nd(1:end) ]; % N(0)=0.
stairs(t,N)

If you decrease the Nperiods, you could see the stairs more precisely.

[Zeynep Toprak]

hello Abdullah, thank you for helphing. I have a question, you know poisson process is different from the poisson distribution. Does this code is totally simulate poisson process by using this formula of f ?

[Abdullah Gül]

The difference between them comes from randomness. I mean when you want to simulate a series of discrete events(e.g. Poisson dist.), its name becomes Poisson process. Here, the timing of events is random, but your model is still Poisson. Hence, you could use that code.

[Zeynep Toprak]

perfect! I'm confusing them most of time. I see now. Many thanks :)

Answer 2

[Sulaymon Eshkabilov]

Here is the alternative solution:

T = 1;
lambda = 12;

t  = linspace(0, T, 500);
k=randi([1,10], 500);
  
f = (lambda.^k) .* exp(-lambda) ./ factorial(k);
N = sum(f);
disp(N);
stairs(t,N)

Good luck

[Zeynep Toprak]

Thank you very much for your help. Is this my second code is true to simulate poisson process? I thought it was trivial, but you confirm it right? and how can I do this the same thing by using poissrnd() ?

[Zeynep Toprak]

And also, it must be N(0) = 0 by the definition of poisson process (Poisson process defintion link), but here, it is not zero.

Answer 3

[Sulaymon Eshkabilov]

Why not to use MATLAB's built in fcn: poissrnd()

[Le Duc Long]

Hi Sulaymon Eshkabilov and Zeyneh Toprak,

I am also interested in this issue. My problem is to get the number of vehicles to an intersection with the Poisson distribution, lambda = 0.5 (vehicle/second). Can you help me code for vehicle traffic (numbers of vehicles) distributed by time period 500 seconds, step of time = 1 second?

Thanks so much!

[Sa'adatu Abubakar]

Hello I'm also interested in the area please can you help me with the MATLAB code for poison arrival and general servive with single server?. Thank you

[Sa'adatu Abubakar]

please I mean poisson arrival with general service and single server.

Answer 4

[Najme Benvari]

Hi I read the Poisson production code when the jump size is random and the intensity of the jumps is Stochastic and a CIR process. I will be very thankful.

[Najme Benvari]

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